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This article suggests that a type can occur more times than it has tokens. e.g., the type mister smith has a single token but occurs twice in the list of lottery winners. The article bases that on sequences

the same person occurs twice in the sequence of New Jersey million dollar lottery winners, remarkably enough. If we think of an expression as a sequence, then the air of mystery over how the same identical thing can occur twice vanishes

Can such a sequence be constructed of tokens that cannot repeat? If not, then it seems that type can't have further types within it

[sequences] distinguish tokens of types from occurrences of types whenever types of things [can] have other types of things occurring in them.

which is exactly what the idea of a "sequence" is meant to resolve.

If lottery winners are barred from playing the lottery again, then the type mister smith can only occur once, in his token. That seems to suggest that each of those lottery winners are tokens of types that do not occur abstractly (lottery winners), but only occur in - and are only - their tokens.

  • According to the article the relationship of a type to token isn't of the form you stated; can you give an example of the kind of type you're theorising about: a type which has only one token, and that not accidentally, but neccesarily? – Mozibur Ullah May 23 '15 at 17:06
  • If there is no such type then it's like examining unicorns that live on Mars; which isn't even a fictional possibility... – Mozibur Ullah May 23 '15 at 17:08
  • "the relationship of a type to token isn't of the form you stated" can you be more specific? anyway my death is a type with one token, if death is irreversible –  May 23 '15 at 18:15
  • At the very beginning of the article you quote; they write 'the distinction between type and token is an ontological one; between a general sort of thing and it's particular concrete instances'; thus to rely on on the condition that you suppose is to constrain it accidentally; it's not a proper part of its sense; one can, as you did, manufacture such types simply by starting from a particular concrete thing, and suppose a type that only refers to that; for example from the concrete teacup that is before me a type that refers only to this; this is similar to Lewises rigid designator except – Mozibur Ullah May 23 '15 at 20:48
  • He uses this to explore names and things in possible worlds; whereas here of course there is only this world. – Mozibur Ullah May 23 '15 at 20:49
  • i kinda get what you may be getting at, but it's still unclear what you think i have misunderstood or misapplied ? regardless of whether "my death" is a type [which i do think it is in a way that "this cup" isn't] –  May 23 '15 at 21:06
  • "we typically flee in the face of death... we regard death as simply more cases or instances of death, as if they were mere tokens of some essentially impersonal type" i.e. if the token my death has a type then it is not the same type as "death" –  May 23 '15 at 21:12
  • Rose is a rose is a rose is a rose. "In one sense of ‘word’ we may count three different words; in another sense we may count ten different words." I an imagine how rose could be 4 different words but how can "is" or "a" be considered as 3 different words? – Ronnie Royston May 26 '15 at 03:28
  • i think the bounty additional note is poor, but i can't edit it! i think... the question is ok as it is, can types that cannot amount to a sequence occur more than once –  Oct 12 '15 at 19:48
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    As types need, to be understood as a type, more than one token, I think that saying "'my death' is a type" just is a wrong use of the term "type". It can be a token of the type "important events in your life" or "the deaths in your family". So every type could be expressed as a sequence consisting of its tokens, there is nothing obscure behind that thought. I think the difference of token and occurance, which is described in the article, is overseen too easily. – Philip Klöcking Oct 12 '15 at 21:11
  • Ok, so consider your death the token of the type of your deaths. You have no doubt mentioned your death many times. Each occurrence is of the same token. So what prevents there being a list? Lists do not 'contain' the things they contain in a sense that makes this problematic, only in a looser sense, involving reference. –  Oct 12 '15 at 21:40
  • For clarity @PhilipKlöcking -- that is 'overlooked' not 'overseen'. An Oversight Committee is not in charge of oversights, except perhaps in the sense of preventing them. –  Oct 12 '15 at 21:49
  • @PhilipKlöcking not sure why you think i've confused occurrence and type, can you elaborate please i don't think i have tbh ! –  Oct 12 '15 at 21:51
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    @jobermark: thanks ;) And: ocurrance and token. Even if your death can occur only once, there can be several, if not infinite possible tokens thinkable. And only in this sense your death can be a type of them. – Philip Klöcking Oct 12 '15 at 22:15
  • @PhilipKlöcking ? how you can imagine annihilation re-occurring what is re-occurring when there's nothing there to be annihilated? we're only talking about annihilation, not corpses or anything –  Oct 13 '15 at 05:47
  • @MATHEMETICIAN: That actually is my point: Although you can only die once (occurance), as long as your death did not occur it can be described as a type including the thinkable possibilities of your death as tokens, with infinite number. BUT as soon as the event realizes it does not make even sense anymore to describe "your death" as a type if you do not take it in an abstract way the like I described (the possibilities thinkable, the actualization as one of the tokens). Apart from that, types need more that one token to be constituted, single tokens are derived from other types like "death" – Philip Klöcking Oct 13 '15 at 08:15
  • @PhilipKlöcking i'm not sure that's your "point". you're right that i can imagine many tokens of this type, but not at once anyway, not i think express it having more than one token –  Oct 13 '15 at 08:19
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    Another point is that "sequence" does not have to mean "sequence in time", as you seem to take it. A sentence is a sequence, as well as of letters as of words. A sequence may be all tokens of a type put in a line (all STRINGs of an ARRAY, as an example out of programming, or all variables of a vector in mathematics - in these examples only the type occurs and the tokens only as part of the type). And as I take the article, having the background of linguistics, it is probable that "sequence" is meant in this broader sense. – Philip Klöcking Oct 13 '15 at 08:39
  • but anyway a sequence of tokens would be of things in time, whether or not we define when they occur. i would imagine but barely care anymore, that we can't express many self annihilations, whether or not their time is left undefined except as a variable –  Oct 13 '15 at 08:46
  • Look at the film Groundhog Day: Of course we can express many of them. I can kill myself by shooting, hanging, jumping... Your point just is not clear. Or to say with Wittgenstein (from memory) "As long as you can't say it, you haven't even thought it right." – Philip Klöcking Oct 13 '15 at 09:07
  • he's not annihilated at all tho, it's a clear term - there's nothing left –  Oct 13 '15 at 09:14
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    You really aren't improving your chances of getting this question answered by cluttering up the comments with rants about downvoting which have been discussed extensively elsewhere already per your request. –  Oct 13 '15 at 10:20
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    @PhilipKlöcking You are missing a distinction in addition to the one between occurrence and token, that is reference. Even a single occurrence can appear in a sequence at multiple places, that does not mean it has multiple instances. Dead is dead, even if we cannot agree what that means (brain death, no breath, etc...) There is only one there, there, even if we continue discussing it forever after, making more and more references. –  Oct 13 '15 at 14:32
  • @Keelan point i guess but this question really does seem to be totally on point, at least with the edits i made –  Oct 14 '15 at 15:22
  • Comments are not a forum. Please take extended discussion to chat – Joseph Weissman Oct 30 '15 at 15:42

2 Answers2

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The Type-Token distinction is a modern counterpart of the old universal-particular one.

In some field, like linguistic, it is a useful tool; in general, it has the same problems of the old one.

If we consider the mathematical sequence of numbers <0,1,0,1> this is an "abstract" entity : the two occurrences of 1 are not two tokens (like two different "handwritings" of the same word "one") but exactly two occurrences of the same (abstract) object 1.

Regarding the fact that :

Even a concrete object can occur more than once in a sequence — the same person occurs twice in the sequence of New Jersey million dollar lottery winners, remarkably enough. If we think of an expression as a sequence, then the air of mystery over how the same identical thing can occur twice vanishes.

it seems to me that there is a little confusion; if we consider a written list of the New Jersey million dollar lottery winners, then we have a sequence of names, and it is correct to say that we can have multiple occurrences of the same name (an "abstract" ?).

But if we try to "build" the sequence of the men that won the New Jersey million dollar lottery we cannot arrange them with a multiple occurrence of e.g. John Smith; there are no possible "tokens" of him to be used twice in the sequence.

Thus, it seems to me, the individual John Smith is a particular, and it has little meaning to speak of it as a type or as having occurrencs.

The name "John Smith", as a word, is a type, and thus it may have multiple tokens or occurrences.

Mauro ALLEGRANZA
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Perhaps this is just the Computer Scientist in me talking, but all of this is sheer grammatical effusion resolved readily by a single mathematical notion -- that of equivalence relation.

First the computer science:

A sequence is not a collection, it has more structure than a collection, and less direct possession of its contents. The things in a list aren't tokens of the listed items, and they aren't even occurrences of the listed items, they are references to occurrences of the listed items.

Your death is the single token of its type, and its single occurrence. But the sequence, since it does not, in itself, allow for counting, makes for irrelevant obfuscation. There can be a sequence which just lists the same occurrence of the same token of a single type over and over again, (e.g. "Me, me, me, me...!")

If there is a list of all the Names of God, they all reference God, who (in the general notion of big-G God) has a single token and occurrence.

Sequences may separate tokens in a type, but they do not properly enumerate them, or we would have to say that the entries in the collection of all the names of God somehow create copies of God.

So while there can only be a single collection of the tokens of a type with a single token, there can be many sequences.

Now the math:

The less direct ownership implicit in a sequence and resolved in a collection avoids contention over ownership that leads into confusion, and ultimately paradox. The issue is that the distinctness that is required to reduce a sequence to a collection is not always present and safe to rely upon.

A collection requires an equivalence relation that needs to be proved consistent, if we are going to enumerate distinct elements correctly.

If what you mean by 'contain' is to have all the same tokens, and potentially then some more, the single-type can clearly contain no other types, if it can be collected.

And finally an answer to the question as asked:

Yes, there are types with just one token. But the notion of sequence has nothing to do with it. You need another, slipperier concept, involving equivalence, to reasonably establish the number of tokens of a type.

(That is why we have classes that are not sets, and why category theories can still handle them, while type and set theories cannot necessarily do so. The essential binding force here is not elements or exemplars, it is the clarity of the identity relation.)

And a (kind of weak) example to indicate why this concept is truly essential:

The type of God is not introduced pointlessly. It is a singleton type that cannot be collected. The definition itself implies uniqueness, but two images of God seldom agree. People constantly choose for God to have different essential contents, and those using one set find that their God must have qualities forbidden the other's God. He does not equal Himself, and there is not an equivalence relation between the named versions of him.

In that, the type of God seems to have clear subtypes. There are essentially Christian-ish perspectives upon God (who loves us), essentially Deist-ish perspectives upon God (who has left us to our own devices), essentially Pantheistic perspectives upon God (who loves us as his aspects love us and hates us as his aspects hate us), etc. One might think that these cannot reference the same thing. But they reference a thing with a single, consistent definition and are derived from that definition.

The dual definition of a contained type from the one determined by tokens, is that a type contains those types that limit the original type via additional constraints. By that notion, these versions of God do seem to be subtypes. In this type, additional constraints somehow break down a single token into multiple types, violating our intuition of what a constraint does, but in a way that still retains meaning.

So we have proof that if a type cannot be collected, even if we all agree it only has one occurrence and one token, it may still have subtypes.

  • Let us continue this discussion in chat. –  Oct 14 '15 at 12:27
  • btw i have access to the article where Wetzell explain the idea of sequence, and it seems that she allows them to consist of just one occurrence, but it's not immediately totally clear - and i lack the patience for anything today :/ –  Oct 14 '15 at 16:43
  • (I'm not your downvote) but please take downvote discussions to meta. People are not required to indicate why they downvoted though it is potentially helpful. If you think you're beginning systematically targeted, (1) there's an algorithm that finds and corrects much of that and (2) you can raise it on meta. And by doing so a moderator can try to look into it – virmaior Oct 21 '15 at 02:33